Do lots and lots of experiments on the specific subject you are interested in. For example Gallileo roled a lot of balls down inclined planes, measuring their speeds a different points.

Try to find some "property" that does not change or changes in some simple way. Gallileo found that the velocity of the balls changed, approximately, as the time, leading to the conclusion that v= at and so a, the "acceleration" was constant. He also conjectured that acceleration of falling balls would be constant (he used inclined planes because the velocity of falling balls was too difficult to measure a number of times), then did modified experiments on falling balls to confirm that acceleration was constant.

Other experiments with collisions between balls moving on a flat plane showed that while velocities and angles could change in complicated ways, certain combinations of those remained constant, leading to the concepts of "momentum" and "kinetic energy".

Do lots and lots of experiments on the specific subject you are interested in. For example Gallileo roled a lot of balls down inclined planes, measuring their speeds a different points.

Try to find some "property" that does not change or changes in some simple way. Gallileo found that the velocity of the balls changed, approximately, as the time, leading to the conclusion that v= at and so a, the "acceleration" was constant. He also conjectured that acceleration of falling balls would be constant (he used inclined planes because the velocity of falling balls was too difficult to measure a number of times), then did modified experiments on falling balls to confirm that acceleration was constant.

Other experiments with collisions between balls moving on a flat plane showed that while velocities and angles could change in complicated ways, certain combinations of those remained constant, leading to the concepts of "momentum" and "kinetic energy".

Some experiments may be impossible, like how does one verify string theory by experimentation? You would need a super collider the size of the milky way powered by energy that would be impossible to create with todays technology. The best you can do is perform indirect experiments with existing technology.

Do lots and lots of experiments on the specific subject you are interested in. For example Gallileo roled a lot of balls down inclined planes, measuring their speeds a different points.

Try to find some "property" that does not change or changes in some simple way. Gallileo found that the velocity of the balls changed, approximately, as the time, leading to the conclusion that v= at and so a, the "acceleration" was constant. He also conjectured that acceleration of falling balls would be constant (he used inclined planes because the velocity of falling balls was too difficult to measure a number of times), then did modified experiments on falling balls to confirm that acceleration was constant.

Other experiments with collisions between balls moving on a flat plane showed that while velocities and angles could change in complicated ways, certain combinations of those remained constant, leading to the concepts of "momentum" and "kinetic energy".

In feynman `s book, the nature of physical law. He said that certain laws can be found by trying to find regularities in the observation as you are saying, but in maxwell case, it was ws by see inconsistence within cases of related equations relatated to electrcit and magnestism. In einstein case, it was by resolving paradox of certain know principle of his days. feynman propose that centain ways is by trying to "guess" the equations(law). How might this actually work? Are there resource for this matter?

Are there systematic methods to finding laws of nature? How do physicists do this?

There are no sure methods of discovery (though empiricism is crucial of course, at least in validating the new hypothesis); new theories are, in their crucial aspects at least, 'free creations of the mind'.

No amount of empirical data gathered (assumed very accurate) can assure a breakthrough in science (think for example at Tycho Brahe's case); there are very few cases when 'inductivist mechanistic approaches' really worked.

Einstein’s for example writes in 'On the method of theoretical physics' (paraphrased):

the experience can suggest us useful mathematical concepts but in any case can we deduce them from it. The experience remains of course the only criterion which validates the utility of a mathematical concept for physics…but in some sense I consider true the fact that pure reason is able to grasp Reality, the dream of the Ancients.”

Karl Popper (1934) too notes that:

My view may be expressed by saying that every discovery contains 'an irrational element', or 'a creative intuition', in Bergson's sense. In a similar way Einstein speaks that "there is no logical path... to these laws. They can only be reached by intuition, based upon something like intellectual love".

Even an inductivist like Carnap (1950) accepts this conclusion (paraphrased):

But in one point the present opinions of most philosophers and scientists seem to agree, that the inductive procedure is not, so to speak, a mechanical procedure prescribed by fixed rules. If for instance a report of of observational result is given...there is no set of fixed rules which would lead automatically to the best hypothesis or even a good one. It is a matter of ingenuity and luck for the scientist to hit upon a suitable hypothesis...I am completely in agreement that an inductive machine, which when fed with observational reports would always furnish a suitable hypothesis, is not possible...

But philosophers of science make a clear distinction between the process of discovery and the process of justification (of scientific theories). Some (basically all scientists and enough many philosophers of science) say that at least at the level of justification we can make a clear cut difference between science and pseudo-science...I disagree with this view...from my perspective all we are justified to say is that we can make a (very very) rough difference between science and pseudo science. There are still good grounds for this position...

Let me describe "the scientific method" as it was explained to me in highschool a long time ago. It is pretty systematic, even though it cannot guarantee discovery.

Step 1: Observation. If you cannot observe anything (not necessarily with your eyes, you can use any sense you want) then you cannot gather information and the method won't work. So you must be able to "look" around. So look at something that piques your curiosity. If you're not curious then don't be a scientist.

Step 2: Hypothesis. This is simply a guess as to what is going on. You dream up any imaginable way in which what you observed at step 1 is as it is. Write down your guess, you will shortly try to show that it is correct. New scientists may guess something pretty wild and far out which is not likely to end up being correct. Experienced scientists may guess something closer to the truth. But there is still no assurance that the hypothesis is right no matter whom it comes from.

Step 3: Experimentation. This is the hardest part. You need to implement a set of repeatable procedures that will put your hypothesis to the test. The idea is that each iteration of these steps aims at objectively confirming or refuting your hypothesis. It can be hard to come up with a good test. Sometimes you have to use a not-so-good test and repeat it many more times than if you have a good one. So you repeat your test over and over. This is why the test must be repeatable: anyone anywhere should be able to do the same thing and confirm your results independently.

Step 4: Theory. This is the conclusion reached from the result of experimentation. If the test at step 3 always ends up the same way no matter who does it, you have a description of certain events along a timeline: that's a theory, a description of a so-called "law" of nature confirmed by repeated experimentation. If the results of your tests were all over the map and you cannot find any pattern then you failed to formulate a theory. Better luck next time.

We assume that the "laws of nature" are the real stuff of course, so theories are descriptions of some part of that. Theories can turn out to be wrong or incomplete. The "laws" themselves are a hidden target that can hardly be irrefutably stated.